Abstract
This paper is devoted to the study of the proximal point algorithm for solving monotone and nonmonotone nonlinear complementarity problems. The proximal point algorithm is to generate a sequence by solving subproblems that are regularizations of the original problem. After given an appropriate criterion for approximate solutions of subproblems by adopting a merit function, the proximal point algorithm is verified to have global and superlinear convergence properties. For the purpose of solving the subproblems efficiently, we introduce a generalized Newton method and show that only one Newton step is eventually needed to obtain a desired approximate solution that approximately satisfies the appropriate criterion under mild conditions. The motivations of this paper are twofold. One is analyzing the proximal point algorithm based on the generalized Fischer-Burmeister function which includes the Fischer-Burmeister function as special case, another one is trying to see if there are relativistic change on numerical performance when we adjust the parameter in the generalized Fischer-Burmeister.
Highlights
In the last decades, people have put a lot of their energy and attention on the complementarity problem due to its various applications in operation research, economics, and engineering, see [16, 18, 30] and references therein
Among all various methods for solving the nonlinear complementarity problem (NCP), we focus on the proximal point algorithm (PPA) in this paper
We look into again the PPA for monotone NCP and P0-NCP studied in [42] and [43], respectively
Summary
People have put a lot of their energy and attention on the complementarity problem due to its various applications in operation research, economics, and engineering, see [16, 18, 30] and references therein. (f ): If Fis strongly monotone with modulus μ and Lischitz continuous with constant L, Ψp(x) provides a global error bound for NCP(F), that is, x − x ≤ B2(L + 1) μ
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